There's something going on with achievements' comments/likes. Notifications show content...

Sometimes achievements do not show a notification. (bug)

Can you check your achievements list for the game? Let me know.

1) Go home, then select your profile

2) Go down to "Achievements"

3) Find the game and check the achievement list.

or

1) Say 'Xbox, go to achievements."

Some achievement notifications were not popping for my friend, however he still unlocked them.

There is not a setting that will prevent the achievements from unlocking. Check his achievement list to see if the achievements are showing unlocked and he is just not receiving the notification when they do. If this is the case you will want to do the following.
From the dashboard go to Settings>Preferences>Notifications and be sure that both "Show Notifications" and "Play Sound" are checked. Also, if his online status is set to busy he will not receive the notifications of the achievements being unlocked, to check
this do the following. Again, from the dashboard go to Settings>Preferences>Online Status and make sure he is set to anything but "Busy". Hope this helps.

Hi JoeOtr

Am so sorry you encountering issues at the moment. Could we have a little more information on what you trying to achieve?? It sounds like something maybe with your activity feed perhaps or commenting/liking other players content.... is that correct??

Yes. Every time I get a notification about a post I've liked or commented on, selecting it redirects me to something completely different. And judging by the chaotic comments on that post, it happens to a lot of other people as well.

Notifications, as mentioned above will cause the achievements not to show up.

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Also, if you have your online status set to "busy", it will temporarily disable notifications, and achievements won't show.